Optimal Dividends for a Two-Dimensional Risk Model with Simultaneous Ruin of Both Branches

We consider the optimal dividend problem in so-called degenerate bivariate risk model under assumption that surplus of one branch may become negative. More specific, we solve stochastic control maximizing discounted dividends until simultaneous ruin both branches an insurance company by showing value function satisfies a certain Hamilton–Jacobi–Bellman (HJB) equation. Further, prove is smallest viscosity solution said HJB equation, satisfying growth conditions. Under some additional assumptions, show strategy lies within subclass all admissible strategies and reduce two-dimensional to one-dimensional one. The results are illustrated numerical example Monte Carlo simulated functions.